Salem Numbers and Growth Series of Some Hyperbolic Graphs

Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the -regular graphs associated to regular tessellations of the hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We also prove that these denominators are essentially irreducible (they have a factor of X + 1 when m 2 mod 4; and when = 3 and m 4 mod 12, for instance, they have a factor of X ² – X + 1). We then derive some regularity properties for the coefficients f _n of the growth series: they satisfy K ⁿ – R < f _n < K ⁿ + R for some constants K, R < 0, < 1.